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## Re: Primes again [really Mordell's equation]

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• ... Thanks for making all that so clear, David. It s reassuring that I had found all their top 9 Elliptic Curves, with up to 24*2 integer points. I have
Message 1 of 199 , Nov 2, 2011
>
> I think I've cracked it:
> http://www.inf.unideb.hu/~pethoe/cikkek/72_S_INTGPZ.pdf
> shows that Mike found their record (with goading from Iago?):
> 54225: 48 in S_0 (no cheating)

Thanks for making all that so clear, David.

It's reassuring that I had found all their "top" 9 Elliptic Curves, with up to 24*2 integer points.

I have extended the investigation to abs(delta) <= 10^7, again for n <= 10^8.

There are exactly 3 delta's with > 24 solutions, namely:-
delta solutions
-3470400 25
-5472225 27
9754975 32 [outright winner!]

Here are the 32 solutions to n^3-m^2=9754975:-
214^3=9800344 213^2=45369
220^3=10648000 945^2=893025
236^3=13144256 1841^2=3389281
266^3=18821096 3011^2=9066121
350^3=42875000 5755^2=33120025
416^3=71991296 7889^2=62236321
475^3=107171875 9870^2=97416900
530^3=148877000 11795^2=139122025
680^3=314432000 17455^2=304677025
700^3=343000000 18255^2=333245025
799^3=510082399 22368^2=500327424
814^3=539353144 23013^2=529598169
1019^3=1058089859 32378^2=1048334884
1435^3=2954987875 54270^2=2945232900
1690^3=4826809000 69405^2=4817054025
1870^3=6539203000 80805^2=6529448025
1880^3=6644672000 81455^2=6634917025
2680^3=19248832000 138705^2=19239077025
3364^3=38068692544 195087^2=38058937569
5026^3=126960157576 356301^2=126950402601
6910^3=329939371000 574395^2=329929616025
8120^3=535387328000 731695^2=535377573025
14450^3=3017196125000 1737005^2=3017186370025
14800^3=3241792000000 1800495^2=3241782245025
19910^3=7892485271000 2809355^2=7892475516025
36815^3=49896997643375 7063780^2=49896987888400
68891^3=326954611071971 18081886^2=326954601316996
89710^3=721975682611000 26869605^2=721975672856025
129530^3=2173257047177000 46618205^2=2173257037422025
168896^3=4817903450587136 69411119^2=4817903440832161
232100^3=12503322161000000 111818255^2=12503322151245025
626866^3=246333878914829896 496320339^2=246333878905074921

I conjecture that, with >=64 integer points, the Mordell Elliptic Curve
y^2=x^3-9754975
will be found to have high rank, say 4 or 5.

Puzzle: compute its rank, torsion subgroup, etc.

Mike
• ... That was 9 months ago. Since then, this learning process by David, Kevin and me has continued, and if you visit that link you will find that those 18
Message 199 of 199 , Sep 1, 2012
>
> > is growing rather slowly
>
> This continues, with merely 18 curves currently in that rank-9 file.
> However, I hope that Mike may soon add to these, since in
> the case y^2 = x^3 + k with k < 0 his systematic methods
> may be more powerful than anything that Kevin or I have contrived.
>
> In any case, this continues to be a learning process for us,
> so thanks again to Cino:
> and then Robin:
> for getting us started.
>
> David
>

That was 9 months ago.
Since then, this "learning process" by David, Kevin and me has continued, and if you visit that link you will find that those 18 elliptic curves have grown somewhat in number! And there are related pages, if you replace "9" by "8" thru "12" in the URL (that signifying the rank of the curves).

We have recently ventured into the rather scary territory of the Tate-Shafarevich group, which is currently not deeply understood by the number theory community. (It is a famous open problem to prove that it is always of finite order.)

You may be interested in our today's post to the NMBRTHRY list:
https://listserv.nodak.edu/cgi-bin/wa.exe?A2=nmbrthry;8f3b553f.1208
which describes a nontrivial result from our researches in this area.

Mike
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